Problem 35 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the graphs of \(f\) and \(g\) in the same coordinate plane. (Include two full periods.) $$\begin{array}{l} f(x)=\cos \pi x \\ g(x)=1+\cos \pi x \end{array}$$

Step-by-Step Solution

Answer

The function \(f(x)=\cos \pi x\) is a standard cosine wave stretching between y = -1 and y = 1 with a period of 2. The function \(g(x)=1+\cos \pi x\) is the same cosine wave but has been shifted upward by one unit.

Step 1: Plotting f(x)

The function \(f(x)=\cos \pi x\) is a cosine function with period change because of the π multiplier to the x. The period will be \(\frac{2\pi}{\pi}=2\). First, plot the key points for one standard period of cosine, which are (0,1), (0.5,0), (1,-1), (1.5,0), and (2,1). These points plot out one full period of the function f. As we need two full periods, repeat these points for the second period, so (2,1), (2.5,0), (3,-1), (3.5,0), and (4,1). Draw a curve through these points.

Step 2: Plotting g(x)

The function \(g(x)=1+\cos \pi x\) is a vertical shift of the function \(f(x)=\cos \pi x\) up 1 unit. So, all the points from function f will be shifted up. Start by moving each point of f up one unit. For the first period, the points from step 1 will shift to (0,2), (0.5,1), (1,0), (1.5,1), and (2,2). For the second period, the points shift to (2,2), (2.5,1), (3,0), (3.5,1), and (4,2). Draw a curve through these points.

Step 3: Finalizing the Graph

We now have two full periods of the functions \(f(x)=\cos \pi x\) and \(g(x)=1+\cos \pi x\) on the same coordinate plane. Make sure the graphs are cleanly drawn, label the significant points and both functions clearly.

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions, often written as \(f(x) = \cos x\). It originates from the unit circle, where it represents the x-coordinate of a point on a circle of radius 1 as the angle moves around the circle. It's a periodic function, meaning it repeats its values in regular intervals or periods. Its standard period is \(2\pi\), meaning after an interval of \(2\pi\), the function starts again from the same point.

• The amplitude, or height of the wave, is always between -1 and 1 for the standard cosine function.
• Its graph starts at (0, 1), drops to (\(\pi\), -1), and returns to (\(2\pi\), 1), indicating one full cycle.
• The maximum points are at the beginning and end of the cycle, and the minimum point is right in the middle.

Understanding the standard behavior of cosine functions helps us when modifications are applied, like period changes and vertical shifts. This foundation is crucial for effectively maneuvering through trigonometric graph transformations.

Period Change

The period of a trigonometric function is the length required for the function to repeat its pattern. For the cosine function, the standard period is \(2\pi\). However, adjustments to the function's formula can alter this period.A period change occurs when the variable inside the cosine function is multiplied by a coefficient. Such a transformation is seen in our exercise where the function \(f(x) = \cos \pi x\) results in a period change:

• The coefficient \(\pi\) compresses the period by the same factor.
• To calculate the adjusted period, use the formula \(\frac{2\pi}{b}\), where \(b\) is the coefficient inside the trigonometric function.
• In our case, \(b = \pi\), so the period is \(\frac{2\pi}{\pi} = 2\).

This means the function's cycle completes in an interval of length 2. Thus, visually it appears more tightly packed compared to the standard cosine, showing how coefficients affect periodicity.

Vertical Shift

A vertical shift moves the entire graph of a function up or down without affecting its shape or horizontal position. This transformation is applied to a function \(y = a + f(x)\) where \(a\) is the amount of shift.In our exercise, the function \(g(x) = 1 + \cos \pi x\) represents a vertical shift:

• The graph of \(f(x) = \cos \pi x\) is shifted vertically upwards by 1 unit.
• This means every y-value on the graph increases by 1.
• If the original graph, \(f(x)\), passed through (0, 1), the new graph, \(g(x)\), will pass through (0, 2).

Vertical shifts are handy for adjusting the position of the function to better fit certain conditions or represent specific data. They are straightforward but impactful changes to any cosine or trigonometric graph.